Researchers have investigated the angular eigenvalue problem of the extreme charged C-metric, a solution to Einstein's equations describing an accelerating, charged black hole. In the extreme limit, where the electric charge Q equals the mass M of the black hole, the governing differential equation simplifies from a Fuchsian equation with five regular singular points into a Confluent Extended Heun Equation. This simplification is key to analytically tackling a complex system that typically requires numerical methods.

To analytically evaluate the angular spectrum, the team formulated a decoupling limit within the dual four-dimensional $\mathcal{N}=2$, $\mathrm{SU(2)}\times \mathrm{SU(2)}$ linear quiver gauge theory. This framework allowed them to derive a "parameter dictionary" and renormalized Matone relations. These relations are crucial because they absorb the macroscopic residue shifts induced by singularity fusion, a phenomenon that occurs when the singular points of the differential equation collapse in the extreme limit.

Based on the regular boundary conditions of the angular equation, the researchers utilized the instanton counting method. This method enabled them to establish an algebraic quantization condition, which in turn yielded the angular eigenvalues. The results obtained through this analytical method are consistent with previous numerical results, validating the theoretical approach. This advancement not only deepens our understanding of the C-metric but also establishes a bridge between black hole solutions and gauge theories, opening new avenues for studying complex gravitational systems through their duality with quantum field theories.